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Suppose $ X $ is a uniformly distribution over $(0,1)$. How to find transformations $Y=g(X)$ to produce random variables with the Poisson distribution?

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You can go for $g:=F^{-1}$ where $F$ denotes the CDF of this Poisson distribution.

In general if $X$ has uniform distribution on $(0,1)$ and $F$ is a CDF of some distribution then it can be shown that random variable $Y:=F^{-1}(X)$ has function $F$ as CDF.

Here $F^{-1}:(0,1)\to\mathbb R$ is prescribed by:$$u\mapsto\inf\{x\in\mathbb R\mid F(x)\geq u\}$$

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